Everyone has brown eyes

Question

I'm going to prove that everyone's eyes are the same color. Ready?

If there is only one person, then it's obviously true; this person's eyes are the same color that this person's eyes.

Suppose it is established that $(n-1)$ persons must have the same eye color. Consider $n$ persons: the $(n-1)$ first have the same eye color, and the $(n-1)$ last have the same eye color. Since the two overlap, everyone has the same eye color.

My initialization is verified, and so is my induction. Since I have brown eyes, everyone has brown eyes. Wait a minute, what?

Answer 1

This works except when $n=2$.

That's why one can say that if any TWO people have eyes of the same color, then everyone's eyes have the same color.

Two commonplace forms of mathematical induction are these:

• The case $n=1$ is trivial, and the hard part is the induction step;
• The case $n=1$ is vacuously true; the induction step is trivial and relies on the case $n=2$ and on the induction hypothesis; and the hard part is the case $n=2$.

Your proof is of the second kind.